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Added by
Luciano Lapas Calheiros
Publication date
2008
fields
Physics
and research's language is
English
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Recent investigations call attention to the dynamics of anomalous diffusion and its connection with basic principles of statistical mechanics. We present here a short review of those ideas and their implications.
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A recent paper [M. H. Lee, Phys. Rev. Lett. 98, 190601 (2007)] has called attention to the fact that irreversibility is a broader concept than ergodicity, and that therefore the Khinchin theorem [A. I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York) 1949] may fail in some systems. In this Letter we show that for all ranges of normal and anomalous diffusion described by a Generalized Langevin Equation the Khinchin theorem holds.
We address this work to investigate some statistical properties of symbolic sequences generated by a numerical procedure in which the symbols are repeated following a power law probability density. In this analysis, we consider that the sum of n symbols represents the position of a particle in erratic movement. This approach revealed a rich diffusive scenario characterized by non-Gaussian distributions and, depending on the power law exponent and also on the procedure used to build the walker, we may have superdiffusion, subdiffusion or usual diffusion. Additionally, we use the continuous-time random walk framework to compare with the numerical data, finding a good agreement. Because of its simplicity and flexibility, this model can be a candidate to describe real systems governed by power laws probabilities densities.
We generalize the reaction-diffusion model A + B -> 0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that linear response breaks down: the average displacement of the reaction front grows as the square-root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.
An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets of both backbones and fingers are constructed on the entire (infinite) $y$ and $x$ axises. To this end we suggested a special algorithm of this special construction. The transport properties of the fractal mesh is studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $langle x^2(t)ranglesim t^{beta}$, where the transport exponent $beta<1$ is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with $beta>1$ has been observed as well when the environment is controlled by means of a memory kernel.
The characterization of diffusion processes is a keystone in our understanding of a variety of physical phenomena. Many of these deviate from Brownian motion, giving rise to anomalous diffusion. Various theoretical models exists nowadays to describe such processes, but their application to experimental setups is often challenging, due to the stochastic nature of the phenomena and the difficulty to harness reliable data. The latter often consists on short and noisy trajectories, which are hard to characterize with usual statistical approaches. In recent years, we have witnessed an impressive effort to bridge theory and experiments by means of supervised machine learning techniques, with astonishing results. In this work, we explore the use of unsupervised methods in anomalous diffusion data. We show that the main diffusion characteristics can be learnt without the need of any labelling of the data. We use such method to discriminate between anomalous diffusion models and extract their physical parameters. Moreover, we explore the feasibility of finding novel types of diffusion, in this case represented by compositions of existing diffusion models. At last, we showcase the use of the method in experimental data and demonstrate its advantages for cases where supervised learning is not applicable.
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